A consensus model for large-scale group decision making with hesitant fuzzy information and changeable clusters
Introduction
Consensus is an essential part of group decision making (GDM) problems. However, a rational consensus procedure is not just a pooling or aggregation of opinions, but a process in which rationally motivated changes are made based on individual preferences [1]. In general, consensus is defined as a dynamic and interactive group decision process coordinated by a moderator. The moderator, who plays a central role in the decision making, provides feedback, advice, and suggestions to assist the decision makers move towards a predefined consensus level. In this paper, the individuals in a group are called participators, which could refer to agents, decision makers, experts, or just people who provide online comments. The moderator determines which consensus model is most suitable and then decides on the parameters for the selected model. Cabrerizo et al. [2] provided an overview of fuzzy consensus models. Recently, a lot of new models have been presented; for iteration-based approaches, the reader can refer to [3], [4], [5], [6], [7], [8], [9], and for optimization-based approaches, the reader can refer to [10], [11], [12], [13].
Previous consensus models have generally only considered a small number of decision makers. However, the rapid development of the economy and technology has increased the demand in organizations. For example, e-democracy and social networks require large-scale group decision management [15], [16]. As emergency events often have a large impact on public interest, emergency management often requires the participation of many decision makers from varying professional backgrounds [17]. Another example is the teacher appointment reformation system in universities, in which all university teachers are required to take part in the decision process [18]. The number of decision makers in the current large-scale GDM (LGDM) problems varies from dozens to thousands. In LGDM problems, because it encounters a vast variety of backgrounds and different resources and information are available for different decision makers, to obtain an agreement among the participators are even more difficult than for usual GDM problems. Therefore, consensus reaching processes is also an essential issue for LGDM.
As LGDM consensus models are beginning to attract greater attention, several new models have been developed. Palomares et al. [16] proposed some schemes to detect and manage individual and subgroup noncooperative behaviors. A monitoring tool based on self-organizing maps was complemented to facilitate a visual analysis of the process performance over time. Palomares et al. [19] further developed a semi-supervised consensus support system to facilitate the discussion process. The novelty of such a system was that the agents were allowed some degree of autonomy and were able to conduct most of the discussion process by themselves, with supervision only being necessary when the agents requested critical changes to their opinions. Pérez et al. [20] presented a consensus model in which the experts’ relevance or importance levels were considered in the feedback mechanism. Quesada et al. [21] introduced a methodology to deal with non-cooperative behaviors, in which a uninorm-based weighting scheme was applied to assign importance weights to the experts. Xu et al. [22] presented an exit-delegation mechanism to deal with clusters; when the proximity degree is not satisfied, the cluster is advised to exit the decision-making process and a delegation mechanism is then adopted to reserve the cluster’s influence by setting trust weights for the other clusters. Xu et al. [17] further considered a consensus model that managed minority opinions and noncooperative behavior. Liu et al. [18] proposed a method to determine decision weights for multiple groups based on subjective and objective information. Dong et al. [23] introduced three kinds of noncooperative behavior. Almost all these studies have focused on noncooperative behavior; when a noncooperative behavior was detected, it was managed by penalizing the importance weights of the identified subgroup. From a different perspective, this paper focuses on representing the clusters’ preferences and achieving a predefined consensus level by changing individual preferences.
There have been significant developments in consensus models for LGDM problems. However, as pointed out in [25], there are still challenges that need to be addressed. The classical consensus models in general are not suitable to be directly used for LGDM problems [24]. One important challenge is in the representation of cluster preferences. Zhu and Xu [39] defined hesitant judgments to describe the hesitancy for a group under the condition that the judgments provided by individuals could not be aggregated or revised. Zhang et al. [25] utilized the linguistic distribution assessments proposed in [26] to represent a group’s linguistic information and proposed an approach to solve multigranular linguistic distribution assessments. Except the methods described in [25], [26], there are other ways to represent cluster preferences. In the above mentioned approaches, the decision makers are assumed to used only a single numerical value or linguistic term to represent their preferences. However, in more complex decision settings, the decision makers may be hesitant about the numerical or linguistic variables needed to express their preferences. Hesitant fuzzy sets (HFSs) were first proposed by Torra [27] to deal with hesitation situations. Since that time, the importance and rationality of using HFS in decision making has been well justified [28], [29]. Rodriguez et al. [30] then proposed a hesitant fuzzy linguistic term set (HFLTS) to deal with linguistic decision making. Both HFS and HFLTS have since then been widely considered [31], [32]. Wu and Xu [33] presented a possibility distribution for a HFLTS. Xu and his team proposed a probabilistic linguistic term set which allowed the sum of the probability information to be less than or equal to one [34]. They then developed various important approaches based on their probabilistic linguistic term sets and hesitant probabilistic fuzzy sets [35], [36], [37], [38]. The methods developed in [26], [33], [34] provided viable alternatives to represent preferences for a group. In line with these previous researches, the first objective of this paper is to utilize the developed representations for LGDM under a fuzzy setting.
Another challenge for LGDM problems is designing a suitable consensus reaching process. As mentioned, most previous studies have focused on managing noncooperative behavior through updating the cluster weights. However, the actual clusters obtained in these studies remain the same throughout the decision process [16], [17], [19]. This is in general not true since the decision makers may change their preferences when the available information changes. Further, methods for modifying individual preferences without changing the cluster preferences have not been addressed. If the cluster preferences are only addressed, then useful information about the participants in that cluster is not considered, which could hinder the broad acceptance of the final decision. Hence, even in a LGDM situation, a feedback mechanism focused on individuals is superior to one based on the clusters. Therefore, the second objective of the paper is to present a new consensus model for LGDM problems that addresses these issues. In comparison with previous consensus approaches for LGDM, the proposed consensus model has some distinctive features:
- (1)
The proposed consensus model is an interactive consensus model. In most existing LGDM models, the consensus process is managed through an automatic approach that updates the weights of the obtained clusters. However, feedback from individuals is very important to make the decision result more acceptable and persuasive. With the proposed feedback strategy, it is easy to identify the alternatives, the pair of clusters, the clusters and the individuals that need to change their preferences.
- (2)
The clusters in the proposed model are allowed to change. As the individuals are able to modify their preferences throughout the consensus reaching process, the clusters may change; therefore, the adoption of fixed clusters is only one special case in the proposed model. Further, with the assistance of the decision support system, the clusters are recognized as virtual clusters; therefore, any change in the clusters does not influence the preference modification of the individuals as the easy to follow rules in the model can be used to guide any modifications.
The remainder of this paper is organized as follows. Section 2 introduces a preference representation for a cluster by typical hesitant fuzzy elements with possibility distributions. Section 3 develops a consensus framework for the LGDM. Section 4 presents the k-means based clustering method, the consensus measures and the feedback strategy for the proposed model in detail. Section 5 gives an example of emergency decision making to illustrate the proposed approach. Concluding remarks are given in Section 6.
Section snippets
Distribution-based concepts and their relationships
This section will start with the definition of possibility distributionbased hesitant fuzzy element (PDHFE). After that, the relationships between various distribution-based concepts are given. The justification of utilizing possibility distribution is also discussed.
Consensus framework for the LGDM
In this section, a framework is designed to solve the consensus reaching process for a LGDM problem.
Proposed consensus model
The consensus reaching process is an iterative, dynamic process that has several discussion rounds. The main phases of the iteration-based model are: gathering preferences, computing the agreement level, consensus control, and feedback generation [60]. In such an iterative process, the moderator is the key figure, as they are responsible for supervising and guiding the decision makers through the discussion process. An implicit rule is that the participating decision makers agree to modify
Illustrative example
This case is based on the emergency decision making problem discussed in [22]. An 7.0 magnitude earthquake hit Ya’an City, a southwest city in Sichuan Province, China, at 8:02 am on April 20, 2013. After the earthquake, the Thirteenth Army of the Chengdu military immediately adopted an emergency response plan based on instructions from the central government. In less than two hours, the air force and naval air force successively arrived at the epicenter to deploy rescue teams. Due to the
Conclusions
In a LGDM problem, making a consensus-based decision has become more important for participants and stakeholders. This paper focused on a consensus model for LGDM problems. The main contributions of the paper are as follows
- (1)
A possibility distribution for typical hesitant fuzzy elements called PDHFE was introduced. Three cases were justified for the use of PDHFE in decision making. This was shown to be especially useful when representing the evaluations in clustered subgroups for LGDM problems in
Acknowledgments
The authors are very thankful to the editors and anonymous reviewers for providing very thoughtful comments which have lead to an improved version of this paper. This work was supported by the National Natural Science Foundation of China (Nos. 71671118, 71301110, 71501137, and 71601134) and also supported by the International Visiting Program for Excellent Young Scholars of Sichuan University and the Fundamental Research Funds for the Central Universities (skqy201525).
References (65)
- et al.
Using consensus and distances between generalized multi-attribute linguistic assessments for group decision-making
Inf. Fus.
(2014) - et al.
A peer-to-peer dynamic adaptive consensus reaching model for the group AHP decision making
Eur. J. Oper. Res.
(2016) - et al.
A group decision making model considering both the additive consistency and group consensus of intuitionistic fuzzy preference relations
Comput. Ind. Eng.
(2016) - et al.
Identifying conflict patterns to reach a consensus – a novel group decision approach
Eur. J. Oper. Res.
(2016) - et al.
Ratio-based similarity analysis and consensus building for group decision making with interval reciprocal preference relations
Appl. Soft Comput.
(2016) Distance-based and ad hoc consensus models in ordinal preference ranking
Eur. J. Oper. Res.
(2006)- et al.
Minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making
Inf. Sci. (NY)
(2015) - et al.
Two consensus models based on the minimum cost and maximum return regarding either all individuals or one individual
Eur. J. Oper. Res.
(2015) - et al.
Mathematical programming methods for consistency and consensus in group decision making with intuitionistic fuzzy preference relations
Knowl. Based Syst.
(2016) - et al.
A concise consensus support model for group decision making with reciprocal preference relations based on deviation measures
Fuzzy Sets Syst.
(2012)
A two-layer weight determination method for complex multi-attribute large-group decision-making experts in a linguistic environment
Inf. Fus.
Consensus model for multi-criteria large-group emergency decision making considering non-cooperative behaviors and minority opinions
Decis. Support Syst.
A method for large group decision-making based on evaluation information provided by participators from multiple groups
Inf. Fus.
Managing experts behavior in large-scale consensus reaching processes with uninorm aggregation operators
Appl. Soft Comput.
A dynamical consensus method based on exit-delegation mechanism for large group emergency decision making
Knowl. Based Syst.
Integrating experts’ weights generated dynamically into the consensus reaching process and its applications in managing non-cooperative behaviors
Decis. Support Syst.
Consistency and consensus measures for linguistic preference relations based on distribution assessments
Inf. Fus.
Multiple criteria decision-making methods with completely unknown weights in hesitant fuzzy linguistic term setting
Knowl. Based Syst.
Consensus-based clustering under hesitant qualitative assessments
Fuzzy Sets Syst.
Probabilistic linguistic term sets in multi-attribute group decision making
Inf. Sci. (NY)
Operations and integrations of probabilistic hesitant fuzzy information in decision making
Inf. Fus.
Consistency-based risk assessment with probabilistic linguistic preference relation
Appl. Soft Comput.
Analytic hierarchy process-hesitant group decision making
Eur. J. Oper. Res.
Distance and similarity measures for hesitant fuzzy sets
Inf. Sci. (NY)
Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms
Inf. Sci. (NY)
Some issues on consistency of fuzzy preference relations
Eur. J. Oper. Res.
Possibility theory, probability and fuzzy sets misunderstandings, bridges and gaps
Fundamentals of Fuzzy Sets
A new version of 2-tuple fuzzy linguistic representation model for computing with words
IEEE Trans. Fuzzy Syst.
Proportional hesitant fuzzy linguistic term set for multiple criteria group decision making
Inf. Sci. (NY)
Survey of clustering algorithms
IEEE Trans. Neural Netw.
Cited by (332)
New consensus reaching process with minimum adjustment and feedback mechanism for large-scale group decision making problems under social trust networks
2024, Engineering Applications of Artificial IntelligenceA robust incomplete large-scale group decision-making model for metaverse metro operations and maintenance
2024, Applied Soft ComputingManaging non-cooperative behaviors in consensus reaching process: A novel multi-stage linguistic LSGDM framework
2024, Expert Systems with ApplicationsAn improved failure mode and effect analysis method for group decision-making in utility tunnels construction project risk evaluation
2024, Reliability Engineering and System SafetyGroup efficiency and individual fairness tradeoff in making wise decisions
2024, Omega (United Kingdom)A minimum cost-maximum consensus jointly driven feedback mechanism under harmonious structure in social network group decision making
2024, Expert Systems with Applications